Chebyshev Polynomials

- Chebyshev polynomials (named after 19th-century Russian mathematician Pafnuty Chebyshev) are two sequences of polynomials, denoted Tₙ(x) and Uₙ(x), that arise in approximation theory, numerical analysis, and engineering. - They are defined recursively (T₀=1, T₁=x, Tₙ₊₁=2xTₙ−Tₙ₋₁) and have the property that among all polynomials of degree n with leading coefficient 1, the Chebyshev polynomial minimizes the maximum absolute error on [−1,1] — making them the foundation of minimax approximation. - Their roots (Chebyshev nodes) are used as interpolation points to minimize Runge's phenomenon (oscillation at interval endpoints), and their use in Chebyshev acceleration (Chebyshev iteration) speeds up convergence of iterative linear system solvers like the conjugate gradient method. - In signal processing, Chebyshev filters (Type I and II) provide steeper roll-off than Butterworth filters at the cost of passband or stopband ripple. In machine learning, Chebyshev polynomials underlie ChebNet, a spectral graph convolutional network that approximates graph convolutions to avoid expensive eigen-decomposition.